First Look at Rigorous Probability Theory 2nd Edition by Jeffrey S Rosenthal ISBN 9812703713 9789812703712

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ISBN 10: 9812703713 
ISBN 13: 9789812703712
Author: Jeffrey S Rosenthal

This textbook is an introduction to probability theory using measure theory. It is designed for graduate students in a variety of fields (mathematics, statistics, economics, management, finance, computer science, and engineering) who require a working knowledge of probability theory that is mathematically precise, but without excessive technicalities. The text provides complete proofs of all the essential introductory results. Nevertheless, the treatment is focused and accessible, with the measure theory and mathematical details presented in terms of intuitive probabilistic concepts, rather than as separate, imposing subjects. In this new edition, many exercises and small additional topics have been added and existing ones expanded. The text strikes an appropriate balance, rigorously developing probability theory while avoiding unnecessary detail.

First Look at Rigorous Probability Theory 2nd Table of contents:

1 The need for measure theory
1.1 Various kinds of random variables
1.2 The uniform distribution and non-measurable sets
1.3 Exercises
1.4 Section summary
2 Probability triples
2.1 Basic definition
2.2 Constructing probability triples
2.3 The Extension Theorem
2.4 Constructing the Uniform[0 1] distribution
2.5 Extensions of the Extension Theorem
2.6 Coin tossing and other measures
2.7 Exercises
2.8 Section summary
3 Further probabilistic foundations
3.1 Random variables
3.2 Independence
3.3 Continuity of probabilities
3.4 Limit events
3.5 Tail fields
3.6 Exercises
3.7 Section summary
4 Expected values
4.1 Simple random variables
4.2 General non-negative random variables
4.3 Arbitrary random variables
4.4 The integration connection
4.5 Exercises
4.6 Section summary
5 Inequalities and convergence
5.1 Various inequalities
5.2 Convergence of random variables
5.3 Laws of large numbers
5.4 Eliminating the moment conditions
5.5 Exercises
5.6 Section summary
6 Distributions of random variables
6.1 Change of variable theorem
6.2 Examples of distributions
6.3 Exercises
6.4 Section summary
7 Stochastic processes and gambling games
7.1 A first existence theorem
7.2 Gambling and gambler’s ruin
7.3 Gambling policies
7.4 Exercises
7.5 Section summary
8 Discrete Markov chains
8.1 A Markov chain existence theorem
8.2 Transience recurrence and irreducibility
8.3 Stationary distributions and convergence
8.4 Existence of stationary distributions
8.5 Exercises
8.6 Section summary
9 More probability theorems
9.1 Limit theorems
9.2 Differentiation of expectation
9.3 Moment generating functions and large deviations
9.4 Fubini’s Theorem and convolution
9.5 Exercises
9.6 Section summary
10 Weak convergence
10.1 Equivalences of weak convergence
10.2 Connections to other convergence
10.3 Exercises
10.4 Section summary
11 Characteristic functions
11.1 The continuity theorem
11.2 The Central Limit Theorem
11.3 Generalisations of the Central Limit Theorem
11.4 Method of moments
11.5 Exercises
11.6 Section summary
12 Decomposition of probability laws
12.1 Lebesgue and Hahn decompositions
12.2 Decomposition with general measures
12.3 Exercises
12.4 Section summary
13 Conditional probability and expectation
13.1 Conditioning on a random variable
13.2 Conditioning on a sub-o-algebra
13.3 Conditional variance
13.4 Exercises
13.5 Section summary
14 Martingales
14.1 Stopping times
14.2 Martingale convergence
14.3 Maximal inequality
14.4 Exercises
14.5 Section summary
15 General stochastic processes
15.1 Kolmogorov Existence Theorem
15.2 Markov chains on general state spaces
15.3 Continuous-time Markov processes
15.4 Brownian motion as a limit
15.5 Existence of Brownian motion
15.6 Diffusions and stochastic integrals
15.7 Ito’s Lemma
15.8 The Black-Scholes equation
15.9 Section summary

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